The error bound for the solution of (REF ) via PGROTP algorithm can be also established.
The next lemma concerning a property of pursuit step is useful in this analysis.
 [1]} 
Let \(x^* \in \mathbb {R}^n\)  be the solution to the problem (REF ) and \(\eta = y-Ax^*\) .
The vector \(u \in \mathbb {R}^n\)  is an arbitrary \(k\) -sparse vector.
Then the optimal solution of the pursuit step
\( z^{*}=\arg \min _{z}\left\lbrace \Vert y-A z\Vert _{2}^{2}: ~ \operatorname{supp}(z) \subseteq \operatorname{supp}(u)\right\rbrace  \) 
